# Convergence of Lyapounov Functions Along Trajectories of Nonexpansive Semigroups: Generic Convergence and Stability

Abstract (Summary)

The main aim of this thesis is to study the convergence of Lyapounov functions
along the trajectories of nonexpansive semigroups in a Hilbert space.
The outline of the thesis is as follows. In Chapter 3, it is shown that a
regularly Lyapounov function for a semigroup of contractions on a Hilbert
space converges to its minimum along the trajectories of the semigroup. In
Chapter 4, we show that while a convex Lyapounov function for a semigroup
of contractions on a Hilbert space may not converge to its minimum along
the trajectories of the semigroup, it converges generically along the trajectories
of the semigroups generated by a class of bounded perturbations of
the semigroup generator. In Chapter 5, we show that the regularly Lyapounov
function nearly converges to its minimum along the trajectories of
the semigroups generated by small bounded perturbations of the semigroup
generator. Besides that we study a problem of interest in its own right, about
the direction of movement of the element of minimal norm in a moving convex
set, in Section 4.9. We show that if C is a nonempty closed convex subset
of a real Hilbert space H, e is a non-zero arbitrary vector in H, and for each
t ? R, z(t) is the closest point in C + te to the origin, then the angle z(t)
makes with e is a decreasing function of t while z(t) ? 0.
Bibliographical Information:

Advisor:Associate Professor Bruce Calvert

School:The University of Auckland / Te Whare Wananga o Tamaki Makaurau

School Location:New Zealand

Source Type:Master's Thesis

Keywords:lyapounov functions generic convergence semigroups

ISBN:

Date of Publication:01/01/2005